We optimize the nucleon-nucleon interaction from chiral effective field theory at next-to-next- to-leading order. The resulting new chiral force NNLOopt yields χ2 ≈ 1 per degree of freedom for laboratory energies below approximately 125 MeV. In the A = 3, 4 nucleon systems, the contributions of three-nucleon forces are smaller than for previous parametrizations of chiral interactions. We use NNLOopt to study properties of key nuclei and neutron matter, and demonstrate that many aspects of nuclear structure can be understood in terms of this nucleon-nucleon interaction, without explicitly invoking three-nucleon forces.